A computer virus is a piece of software that can 'infect' a computer, install itself, and copy itself to other computers, without the user's knowledge or permission. It usually attaches itself to other computer programs, datafile, or the boot sector of a Hard drive.
Malware is short for malicious software. Malware is the name that is given to any type of software that could harm a computer system, interfere with and gather a user's data, or make the computer perform actions without the owner's knowledge or permission.
A type of malware that uses malicious code to install software that seems ok, but is hidden to create back doors into a system. This typically causes the loss or theft of data from an external source.
Unlike a virus, a worm is a standalone piece of malicious software that replicates itself in order to spread to other computers. It often uses a computer network to spread itself, relying on security flaws in the target system to allow access.
Spyware is software that aids in gathering information about a person or organization without their knowledge. Spyware can monitor and log the activity that is performed on a target system, like logging keystrokes or gather credit card and other information.
Adware is software that can automatically cause pop-up and banner adverts to be displayed to generate revenue for its author or publisher. A lot of freeware uses Adware but not always in a malicious way. If it was malicious, it would then be classed as spyware or malware.
For more information please see this article - Protect your computer or remove virus, spyware, or other malware in Microsoft Windows.